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Earnings per Share - Schedule of Calculation of Numerator and Denominator in Earnings Per Share (Details) - USD ($)
$ / shares in Units, $ in Millions
 12 Months Ended
Sep. 30, 2024
Sep. 30, 2023
Sep. 30, 2022
Numerator:      
Net income $ 260.8 $ 238.5 $ 207.1
Less: Allocation to participating securities (9.1) (8.1) (6.1)
Net income allocated to common stockholders, basic 251.7 230.4 201.0
Net income allocated to common stockholders, diluted $ 251.7 $ 230.4 $ 201.0
Denominator:      
Common shares outstanding (in shares) 30,539,237 29,936,000 29,355,605
Dilutive potential common shares outstanding:      
Share-based awards (in shares) 1,085,792 993,011 745,706
Diluted shares outstanding (in shares) 31,625,029 30,929,011 30,101,311
Earnings per share:      
Earnings per share - basic (in dollar per share) $ 8.24 $ 7.71 $ 6.85
Earnings per share - diluted (in dollar per share) $ 7.96 $ 7.45 $ 6.67